| L(s) = 1 | + 3-s − 7-s + 9-s + 2·11-s − 5·13-s − 3·17-s + 8·19-s − 21-s + 23-s + 27-s + 29-s − 9·31-s + 2·33-s − 2·37-s − 5·39-s + 7·41-s − 7·43-s − 12·47-s + 49-s − 3·51-s − 13·53-s + 8·57-s − 15·59-s + 7·61-s − 63-s + 4·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.38·13-s − 0.727·17-s + 1.83·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.185·29-s − 1.61·31-s + 0.348·33-s − 0.328·37-s − 0.800·39-s + 1.09·41-s − 1.06·43-s − 1.75·47-s + 1/7·49-s − 0.420·51-s − 1.78·53-s + 1.05·57-s − 1.95·59-s + 0.896·61-s − 0.125·63-s + 0.488·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.48803636929076159751993688364, −6.88134433689583717184030323570, −6.21915775473920136395200462717, −5.14873794794681222185270248428, −4.76684310654483403598195919122, −3.66059814220381632282184188173, −3.16449924081650148395659876770, −2.27336787409042057757248794454, −1.38069361906508789000182694915, 0,
1.38069361906508789000182694915, 2.27336787409042057757248794454, 3.16449924081650148395659876770, 3.66059814220381632282184188173, 4.76684310654483403598195919122, 5.14873794794681222185270248428, 6.21915775473920136395200462717, 6.88134433689583717184030323570, 7.48803636929076159751993688364
